Directional prepositions as numeral modifiers

Dominique Blok

Utrecht University

D.Blok@uu.nl

48th Annual Meeting of the Societas Linguistica Europae

Leiden University Centre for Linguistics

2 September 2015

1 Introduction

Directional numeral modifiers: expressions that can be used as directional prepositionsand as numeral modifiers.

(1) a. Mary walked (all the way) up to the counter.b. You can make up to five copies.

I asked speakers of 15 di↵erent languages about the properties of directional numeralmodifiers (DNMs) in their language. It turns out that DNMs crosslinguistically sharea bundle of properties.

Languages

• Upper-bounded directional numeral modifiers share certain properties in at leastthe following languages:

– Danish

– Dutch

– English

– Farsi

– French

– German

– Greek

– Hebrew

– Hungarian1

– Italian

– Polish

– Romanian

– Russian

– Spanish

– Turkish

• For details, see Blok (2013, 2015).

(2) Polish: do

a. JanJohn

idziegoes

dodo

sklepu.the store.

1The Hungarian expression I studied is kozel. It behaves like a DNM in most ways but seems to have a strongerproximal component than other DNMs and is infelicitous with low numbers.

1

‘John goes up to the store.’b. Dozwolone

Alloweddodo

pieciufive

sztukitems

bagazu.of luggage.

‘It is allowed to take up to five items of luggage.’

(3) Greek: mehri

a. PerpatisameWe walked

mehrimehri

tinthe

akriedge

tis limnis.of the lake.

‘We walked up to the edge of the lake.’b. Ston

In theanelkistiraelevator

khoranefit

mehrimehri

55atoma.people.

‘Up to 5 people can fit in the elevator.’

Main claims:

• In any language, if an upper-bounded numeral modifier is directional, it has thefollowing five properties:

– Its upper bound is cancellable.

– Its lower bound is not cancellable.

– It displays the bottom-of-the-scale-e↵ect.

– It is not clearly downward monotone.

– It does not license NPIs.

• The root of all these properties is that the lower bound of directional numeralmodifiers is asserted while their upper bound is implicated.

Structure of the talk:

• The A/B distinction

• Schwarz, Buccola, and Hamilton’s (2012) ideas about up to

• The bounds of DNMs

• An implicature-based account

• Conclusion

2 Nouwen’s (2010) A/B distinction

Nouwen: numeral modifiers can be categorised into two classes: those that obligatorilygive rise to ignorance e↵ects and those that do not give rise to these e↵ects.

(4) I know exactly how much memory my laptop has...

a. ...and it is {#at most / # at least / # up to} 2GB.

2

b. ...and it is {more than / less than / under} 2GB.

Class A Class BLower bound more than n at least n

over n minimally nfrom n (up)n or more

Lower and between n and n from n to nupper boundUpper bound fewer than n at most n

less than n maximally nunder n up to n

n or fewern or less

Table 1: Classification of numeral modifiers in English

3 Schwarz et al. (2012) on up to

Schwarz et al.: up to is di↵erent from other upper-bounded class B numeral modifierssuch as at most in the following ways:

• It displays the bottom-of-the-scale e↵ect

• It is non-monotone

• It does not license NPIs

The bottom-of-the-scale e↵ect

Schwarz et al.: up to is incompatible with the numeral at the bottom of the scale itquantifies over.

(5) a. At most ten people died in the crash.b. At most one person died in the crash.

(6) a. Up to ten people died in the crash.b. #Up to one person died in the crash.

The bottom-of-the-scale element can be higher or lower than 1:

(7) Context: eggs are sold in cartons of six

a. He bought at most six eggs.b. #He bought up to six eggs.

(8) Context: cakes are sold per slice

a. She bought at most one whole cake.

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b. She bought up to one whole cake.

In all languages I looked at, directional numeral modifiers display the bottom-of-the-scale e↵ect.

(9) Danish:

a. HøjestAt most

enone

personperson

dødedied

vedin

sammenstødet.the accident.

b. #Op tilUp to

enone

personperson

dødedied

vedin

sammenstødet.the accident.

(10) Spanish:

a. Como muchoAt most

unaone

personaperson

muriodied

enin

elthe

accidente.accident.

b. #HastaUp to

unaone

personaperson

muriodied

enin

elthe

accidente.accident.

The bottom-of-the-scale e↵ect is not just a property of up to but of DNMs crosslin-guistically.

NPI licensing

Schwarz et al.: up to does not license NPIs

(11) a. At most three people had ever been in this cave.b. *Up to three people had ever been in this cave.

(12) a. At most three students give a damn about Pavarotti.b. *Up to three students give a damn about Pavarotti.

Again, this appears to be a property of directional numeral modifiers crosslinguisticallyrather than an idiosyncrasy of the English expression up to.

(13) Dutch:

a. ErThere

hoevenmust

maximaalmaximally

vijffive

studentenstudents

teto

komen.come.

‘At most five students have to show up.’b. *Er

Therehoevenmust

totup to

vijffive

studentenstudents

teto

komen.come.

‘Up to five students have to show up.’

(14) French:

a. ?TroisThree

personnespersons

au plusmaximally

onthave

vuseen

qui que ce soit.anyone.

‘At most three people have seen anyone.’b. *Jusqu’a

Up totroisthree

personnespersons

onthave

vuseen

qui que ce soit.anyone.

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If we follow Ladusaw (1979), this suggests that DNMs are either upward entailing ornon-monotone.

Monotonicity

Schwarz et al.: up to is non-monotone.2

(15) a. At most three students smoke. |=b. At most three students smoke cigars.

(16) a. Up to three students smoke. 6|= (Schwarz et al.) / ? |= (my informants)b. Up to three students smoke cigars.

My informants rejected neither the entailment pattern in (16) nor the opposite pattern((16-b) |= (16-a)) in their languages.

Schwarz et al.’s account

Schwarz et al. propose a non-monotone semantics for up to. Their semantics for upto has two components:

1. It sets an upper bound.

2. It contains a range requirement.

(17) Up to ten people died in the crash.

(17) is then taken to mean 1) that according to the epistemic possibilities consideredby the speaker, the maximal number of people who died is ten, and 2) that the numberof epistemic possibilities considered by the speaker must be at least two.

The range requirement gives rise to ignorance e↵ects and accounts for the bottom-of-the-scale e↵ect.

(18) #Up to one person died in the crash.

(18) is ruled out because the range requirement is not satisfied; the only possibilityconsidered by the speaker is the possibility that one person died.

Two issues:

1. The bottom-of-the-scale e↵ect is not fully accounted for. As the 0-possibility isnot ruled out, up to + BOTS numeral still expresses two possibilities: the 0-possibility and the BOTS numeral-possibility. There is no violation of the rangerequirement.

2Schwarz et al.’s explanation of this intuition: ‘Our intuitions indicate that (16-b) cannot be inferred from (16-a).Specifically, [...] in a scenario where the speaker is sure that exactly one, two or three students smoke, while also beingsure that exactly one or two (but not three) students smoke cigars, (16-a) is true and appropriate, while (16-b) is not.’(Schwarz et al., 2012, p.7)

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2. A consequence of positing the range requirement for up to only means you missa generalisation when it comes to accounting for ignorance e↵ects of class Bmodifiers.

4 It’s all in the bounds

There are two additional contrasts between DNMs one the one hand and other upper-bounded class B numeral modifiers on the other hand:

• The lower bound of DNMs is strong and cannot be cancelled.

• The upper bound of DNMs is weak and can be cancelled.

Lower bound

(19) a. At most three students will show up to the lecture, if any.b. #Up to three students will show up to the lecture, if any.

(20) Italian:

a. CiThere

sarannowill be

al massimomaximally

cinquefive

studentistudents

alat the

seminario,seminar,

seif

nonnot

nessuno.none.‘There will be maximally five students at the seminar, if any.’

b. #CiThere

sarannowill be

fino aup to

cinquevife

studentistudents

alat the

seminario,seminar,

seif

nonnot

nessuno.none.

‘There will be up to five students at the seminar, if any.’

(21) Russian:

a. NaAt

seminareseminar

budetwill be

maksimummaximally

5five

studentov,students,

esliif

tamthere

voodbsheat all

budutwill be

studenti.students.

‘There will be maximally five students at the seminar, if any.’b. #Na

Atseminareseminar

budetwill be

doup to

5five

studentov,students,

esliif

tamthere

voodbsheat all

budutwill be

studenti.students.‘There will be up to five students at the seminar, if any.’

Upper bound

(22) a. #Leftovers keep in the refrigerator for at most one week or more.b. Leftovers keep in the refrigerator for up to one week or more.3

(23) a. #At most ten people died in the crash, perhaps even more.

3Source: http://minimalistbaker.com/best-ever-5-minute-microwave-hummus/, last consulted 03-11-2014

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b. Up to ten people died in the crash, perhaps even more.

(24) Romanian:

a. Pana laUp to

trezecithirty

deof

persoanepersons

auhave

venitcome

la petrecere.to the party.

‘Up to thirty people showed up at the party.’b. De fapt,

In fact,credI think

cathat

auhave

venitcome

treizeci si douathirty-two

deof

persoane.persons.

‘In fact, I think that thirty-two people showed up.’

(25) a. Cel multAt most

trezecithirty

deof

persoanepersons

auhave

venitcome

la petrecere.to the party.

‘At most thirty people showed up at the party.’b. ?De fapt,

In fact,credI think

cathat

auhave

venitcome

treizeci si douathirty-two

deof

persoane.persons.

‘In fact, I think that thirty-two people showed up.’

(26) Turkish:

a. PartiyeTo party

3030

kadarup to

insanperson

geldi.came.

‘Up to thirty people showed up at the party.’b. Sanirim

I think3232

insanperson

geldi.came.

‘I think 32 people showed up.’

(27) a. PartiyeTo party

en cokat most

3030

insanperson

geldi.came.

‘At most thirty people showed up at the party.’b. ?Sanirim

I think3232

insanperson

geldi.came.

‘I think 32 people showed up.’

The fact that these properties hold for DNMs crosslinguistically makes sense giventhe fact that in spatial and temporal contexts, these expressions also have a defeasibleend-point.

(28) Joan worked (from 9am) until 10pm today. She may have even stayed laterthan that.

(29) Harry ran (from school) all the way up to his house. I think he may even havegone on to run to the football field after that.

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Contrasts between DNMs and other upper-bounded class B modifiers:

DNMs Other upper-bounded class BBottom-of-the-scale e↵ect Yes NoNPI licensing No YesMonotonicity ? Downward monotoneLower bound Not cancellable CancellableUpper bound Cancellable Not cancellable

Table 2: Summary of the data

5 An implicature-based account

5.1 Two generalisations

I propose that the following two generalisations explain the contrasts between DNMsand other upper-bounded class B numeral modifiers:

1. The lower bound of DNMs is asserted while their upper bound is implicated.

2. All class B numeral modifiers require quantification over a range of values.

In my account, DNMs convey that the degree predicate holds for an interval on ascale consisting of at least two numbers. The asserted lower bound is contextuallydetermined. There is no maximality operator or other mechanism to set an upperbound in the semantics.

(30) Up to ten people died in the crash.

Thus, (30) conveys the possibilities that [1,...,10] people died in the crash withoutexcluding any other possibilities.

The bottom-of-the-scale e↵ect

The bottom-of-the-scale e↵ect can be accounted for in the same way as in Schwarz etal. (2012): using the bottom-of-the-scale numeral leads to a singleton set of possibili-ties, which violates the range requirement.

As the current account comprises a range requirement for all class B numeral modifiers,it predicts that they all display the bottom-of-the-scale e↵ect. This prediction is borneout. The di↵erence between DNMs and other upper-bounded numeral modifiers is thatthe former assert a lower bound.

(31) a. #Up to one person died in the crash.b. #At most zero people died in the crash.

(32) a. Up to two people died in the crash.b. At most one person died in the crash.

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Ignorance e↵ects

Positing a range requirement for all class B numeral modifiers also gives you a uniformaccount of ignorance e↵ects.

NPIs and monotonicity

As DNMs assert a lower bound but not an upper bound, they are predicted to beupward entailing, i.e. (33-a) entails (33-b).

(33) a. Up to three students smoke cigars. |=b. Up to three students smoke.

However, there is a pragmatically derived upper bound. The fact that there is a se-mantic lower bound but a pragmatic upper bound can blur intuitions on entailment.This explains the observed variation in judgments.

The fact that DNMs are semantically upward entailing is compatible with the factthat they do not license NPIs.

5.2 Additional evidence: the interaction with evaluative

adverbs

Evaluative adverbs generally seem to target the assertion of an utterance and not theimplicature, as demonstrated in (34).

(34) a. Fortunately, some students attended the wedding. (Nouwen, 2006)b. Fortunately, the soup is warm.

The speaker of (34-a) is happy that at least some students attended the wedding, notthat not all students did. Similarly, (34-b) is used to convey that it is a good thingthat the soup is at least warm, not that it is not hot.

Similarly, while the speaker of (35-a) expresses her joy about the high number of gueststhat will attend the wedding, the person uttering (35-b) conveys that she is happy thatno more than 100 people will be there. This is evidence for the claim that the assertedcontent of (35-a) is a lower bound while the asserted content of (35-b) is an upperbound.

(35) a. Fortunately, up to 100 people will attend my wedding.b. Fortunately, at most 100 people will attend my wedding.

Again, this property holds for DNMs crosslinguistically, as illustrated below for Farsiand German.

(36) Farsi:

a. KhoshbakhtaneFortunately

mitoonamI can

taup to

5five

roozdays

morakhasiget time

begiram.o↵ work.

9

b. ?KhoshbakhtaneFortunately

unthat

khanandeyesinger

eftezahhorrible

taup to

55(ta)(up to)

ahangsongs

mixanad.sing.‘Fortunately, that horrible singer will sing up to five songs.’

(37) a. ?KhoshbakhtaneFortunately

mitoonamI can

hade aksarat most

panjfive

roozdays

morakhasiget time

begiram.o↵ work.

b. KhoshbakhtaneFortunately

unthat

khanandeyesinger

eftezahhorrible

hade aksarat most

55tato

ahangsongs

mixanad.sing.‘Fortunately, that horrible singer will sing at most five songs.’

(38) German:

a. GlucklicherweiseFortunately

kanncan

ichI

bis zuup to

funffive

Tagedays

freio↵

kriegen.get.

‘Fortunately, I can get up to five days o↵.’b. ?Glucklicherweise

Fortunatelysingtsings

dieserthat

schlechtebad

Sangersinger

bis zuup to

funffive

Songs.songs.

‘Fortunately, that bad singer will sing up to five songs.’

(39) a. ?GlucklicherweiseFortunately

kanncan

ichI

maximalmaximally

funffive

Tagedays

freio↵

kriegen.get.

‘Fortunately, I can get at most five days o↵.’b. Glucklicherweise

Fortunatelysingtsings

dieserthat

schlechtebad

Sangersinger

maximalmaximally

funffive

Songs.songs.

‘Fortunately, that bad singer will sing at most five songs.’

6 Conclusion

Directional numeral modifiers are crosslinguistically di↵erent from non-directional nu-meral modifiers in that their upper bound is cancellable while their lower bound isnot. Assuming that the former is an implicature while the latter is entailed leads toan account of the bottom-of-the-scale e↵ect, monotonicity properties and interactionswith evaluative adverbs.

7 Acknowledgements

I thank Rick Nouwen, Yaron McNabb, Stavroula Alexandropoulou, Elizabeth Cop-pock, Benjamin Spector, Lisa Bylinina, Hanna de Vries, Jeremy Kuhn, Marlijn Meijer,Annemarie van Dooren, and the audience at the Proportions and Quantities workshopat DGfS 2015 for helpful comments and feedback. Any remaining mistakes are myown.

I thank my informants for their judgments: Pernille Borregaard, Anna FolkeLarsen, Annemarie van Dooren, Indira Wakelkamp, Marjolein Wennekers, Marlijn

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Meijer, Lea ter Meulen, Chantal van Dijk, Jeremy Kuhn, Brian Buccola, Ruth Brill-man, Aron Hirsch, Heidi Klockmann, Natalie Lowell, Sara Chan, Claire Marie Dunne,Emma Walker, Fafa Same, Farhad Mirdamadi, Anne-France Pinget, Loıc de Jong,Margot Colinet, Benjamin Storme, Edwige Dugas, Ben Rodenhauser, Anja Gold-schmidt, Stavroula Alexandropoulou, Vicky Stefou, Dimitra Lazaridou-Chatzigoga,Maria Koutiva, Yaron McNabb, Lilla Magyar, Ivano Ciardelli, Vincenzo Tabacco, Na-talia Davidson, Barbara Tomaszewicz, Silvia Radulescu, Ileana Grama, Lisa Bylinina,Anna Volkova, Lena Karvovskaya, Jenna Baranov, Rocıo Romero Merida, EduardoDominiccini, Alexia Guerra Rivera, Isa Bayirili, Wataru Uegaki, Ana Lucia Pessottodos Santos, Bambang Kartono, Mengru Han, Yipu Wei, Yuning Sun, Ishani Guha andher friends Bidisha and Santosh Kumar, and Simona Bernataviciute.

This work was supported by an ERC Consolidator Grant.

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References

Blok, D. (2013). Directional prepositions as numeral modifiers.Unpublished master’s thesis, Utrecht University, available at:http://www.dominiqueblok.org/work.

Blok, D. (2015). The semantics and pragmatics of directional numeral modifiers.Unpublished manuscript, available at: http://www.dominiqueblok.org/work.

Chierchia, G., Fox, D., & Spector, B. (2009). Hurford’s constraint and the theory ofscalar implicatures. In P. Egre & G. Magri (Eds.), Presuppositions and implica-tures. proceedings of the mit-paris workshop (Vol. 60). MIT Working Papers inLinguistics.

Chierchia, G., Fox, D., & Spector, B. (in press). The grammatical view of scalar impli-catures and the relationship between semantics and pragmatics. In P. Portner,C. Maienborn, & K. von Heusinger (Eds.), An international handbook of naturallanguage meaning. Berlin: Mouton de Gruyter.

Ciardelli, I., Groenendijk, J., & Roelofsen, F. (2009). Attention! Might in inquisitivesemantics. In S. Ito, E. Cormany, & D. Lutz (Eds.), Proceedings of SALT 19.

Ciardelli, I., Groenendijk, J., & Roelofsen, F. (2012). Inquisitive semantics. NASSLLI2012 lecture notes.

Coppock, E., & Brochhagen, T. (2013). Raising and resolving issues with scalarmodifiers. Semantics & Pragmatics , 6 (3), 1-57.

Davies, M. (2008). The corpus of contemporary American English (COCA). 450million words, 1990-2012. Retrieved May 2015, from www.americancorpus.org

Grice, P. (1975). Logic and conversation. In P. Cole & J. Morgan (Eds.), Syntax andsemantics 3: Speech acts (p. 41-58). New York: Academic Press.

Hackl, M. (2000). Comparative quantifiers. PhD Thesis, Department of Linguisticsand Philosophy, Massachusetts Institute of Technology.

Ladusaw, W. (1979). Polarity sensitivity as inherent scope relations. PhD Thesis.University of Texas at Austin.

Nouwen, R. (2006). Remarks on the polar orientation of almost. In J. van de Weijer& B. Los (Eds.), Linguistics in the Netherlands 23 (p. 162-173). Amsterdam:AvT Publications, John Benjamins Publishing Company.

Nouwen, R. (2010a). Two kinds of modified numerals. Semantics & Pragmatics , 3 (3),1-41.

Nouwen, R. (2010b). What’s in a quantifier? In M. Everaert, T. Lentz, H. Mulder,Ø. Nilsen, & A. Zondervan (Eds.), The linguistics enterprise. Amsterdam: JohnBenjamins Publishing Company.

Schwarz, B., Buccola, B., & Hamilton, M. (2012). Two types of class B numeralmodifiers: A reply to Nouwen 2010. Semantics & Pragmatics , 5 (1), 1-25.

Spector, B. (2014). Global positive polarity items and obligatory exhaustivity. Se-mantics & Pragmatics , 7 (11), 1-61.

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Appendix

1. Open issues

Proximity

Like almost but unlike at most, DNMs seem to have a proximal component:

(40) a. Fortunately, almost all my friends will attend my wedding.b. Fortunately, up to 200 of my friends will attend my wedding.c. Fortunately, at most 200 of my friends will attend my wedding.

Both (40-a) and (40-b) but not (40-c) suggest the precise number is under but closeto all my friends/200 friends. What is the nature of this element of the meaning ofDNMs?

Directivity

DNMs seem to presuppose that the number they modify is a high number (a phe-nomenon referred to as directivity in Nouwen, 2010b):

(41) [In the context of a commercial]

a. Discounts of up to 50%!b. #Discounts of at most 50%!

What is the nature of this element of their meaning? Is it related to the proximalcomponent?

DE contexts

Clear contrast in the scope of negation:

(42) a. I don’t think there will be discounts of up to 70%.! The highest discount is lower than 70%.

b. I don’t think there will be discounts of at most 70%.! The highest discount is higher than 70%.

While (42-a) conveys that the highest discount is lower than 70%, (42-b) means thatthe highest discount is higher than 70%.4 This is expected if we take up to to conveythat the degree predicate holds for a range of numbers on a scale, while at most ex-presses an upper bound. (42-a) thus means that it is not the case that for all numberson a scale from 1 to 70, there will be discounts of that amount. (42-b) means that themaximum discount is not 70. Negating a maximum is equivalent to expressing highernumbers are among the possibilities.

Less clear contrast in the antecedent of a conditional:4The 14 speakers I asked all agreed with this judgement. These were speakers of Farsi, French, German, Italian,

Polish, Romanian, Russian, Spanish, and Turkish.

13

(43) a. If you received { up to / at most} ten books you did something wrong.b. If you order { up to / at most} ten books you have to pay a delivery fee.c. If you order { up to / at most} ten books you get a discount on your next

purchase.

(43-a) (43-b) (43-c) TotalNo contrast; UB 3 23.1% 8 61.5% 5 38.5% 16 41.0%No contrast; no UB 6 46.2% 1 7.7% 0 0% 7 17.9%Contrast: no UB for DNM 4 30.8% 2 15.4% 8 61.5% 14 35.9%Contrast: no UB for at most 0 0% 2 15.4% 0 0% 2 5.1%

Total 13 100% 13 100% 13 100% 39

Table 3: Judgments on upper bound implicatures in DE environments5

Possible explanation: local implicatures (Chierchia, Fox, & Spector, 2009, in press;Spector, 2014).

(44) a. Joe didn’t see Mary or Sue; he saw both.b. It is not just that you can write a reply. You must.c. I dont expect that some students will do well, I expect that all students

will.

Ignorance e↵ects

Ignorance readings of class B modifiers generally become optional but not absent whenthey occur with a modal or a plural, as in (45).

(45) a. Computers of this kind have at most 2GB of memory.b. John is allowed to bring at most 10 friends.

The sentences in (45) also have a reading where the computers all have the samememory capacity and John is allowed to bring a fixed number of friends, but thespeaker does not know what the exact number is. These ignorance readings in contextswith plurals or modals seem less prominent, if not absent, when a DNM is used, as in(46).

(46) a. Computers of this kind have up to 2GB of memory.b. John is allowed to bring up to 10 friends.

2. The account in inquisitive semantics

Inquisitive semantics

The ideas presented above can be formalised in the framework of inquisitive semantics(e.g. Ciardelli, Groenendijk, & Roelofsen, 2009, 2012), akin to Coppock and Brochha-gen (2013). Inquisitive semantics di↵ers from classical semantics in the following way:

5Judgments from speakers of Dutch, French, German, Greek, Italian, Romanian, Russian, Spanish, and Turkish.

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• In inquisitive semantics, a proposition expresses a set of possibilities. A possi-bility is a set of worlds (or classical proposition). A proposition thus conveys aset of sets of worlds.

• This allows for a richer notion of meaning: two propositions that comprise thesame set of worlds can di↵er in meaning because the structure of the propositionsis di↵erent.

Denotation

I propose the semantics in (47) for DNMs.

(47) Jup toK = {�n�P.f{P (m)| s m n}|f is a choice function}where s > 0 and s 6= n

Using this definition, the semantics of (48) is as in (54-c).

(48) Up to ten people died in the crash.

(49) {f{�w9x[#x = m ^ people(x)(w) ^ died-in-the-crash(x)(w)] |s m 10} | f is a choice function}= {�w9x[#x = m ^ people(x)(w) ^ died-in-the-crash(x)(w)] | s m 10}

= {p1,p2,p3, ...,p10}where pn = {wn, wn+1, wn+2, ...1}

Ignorance e↵ects

Ignorance e↵ects come about through the Maxim of Interactive Sincerity (Coppock &Brochhagen, 2013):

(50) If ' is interactive, then ' is interactive in the speaker’s information set

(51) ' is interactive i↵ J'K contains more than one possibility

As a result of the range requirement, every proposition with a DNM is interactive.Thus, the range requirement in combination with the Maxim of Interactive Sinceritygenerates ignorance e↵ects.

Upper-bound implicature

The structure of (54-c) enables us to derive the upper bound implicature using Coppockand Brochhagen’s exhaustification procedure, given in (52).

(52) exh(P,s ) = {p� q | p 2 P ^ q = {w | 9q0 2s [w 2 q

0 ^ p 6✓ q

0]}}where P is the proposition and s is the question under discussion

This results in the following outcome for (48):

(53) P = {p1, p2, ..., p10} (= {{w1, w2, w3, ...}, {w2, w3, w4, ...}, ..., {w10, w11, w12, ...}})s = {q00, q01, q02, q03, q04, ...} (= {{w0, w1, w2, ...}, {w1, w2, w3, ...}, ...})

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exh(P ,s ) = p1 � q = p1 � {w2, w3, w4, ...} = {w1}. p2 � q = p2 � {w3, w4, w5, ...} = {w2}. .... p10 � q = p10 � {w10, w11, w12, ...} = {w10}

. = {{w1}, {w2}, ..., {w10}}

As (53) illustrates, the exhaustivity operator removes all worlds above w10 from theinformational content, resulting in an implicated upper bound of 10.

Bottom-of-the-scale e↵ect

As the lower bound s of the scale cannot be zero, the existence of at least one elementfor which the predicate holds is entailed. As shown in (54), the bottom-of-the-scalee↵ect is accounted for.

(54) a. #Up to one person died in the crash.b. [up to 1 [ ⌃ [ �n [n-many people died in the crash ] ] ] ]c. {f{�w9x[#x = m ^ people(x)(w) ^ died-in-the-crash(x)(w)] |

1 m 1} | f is a choice function}= {�w9x[#x = m ^ people(x)(w) ^ died-in-the-crash(x)(w)] | 1 m 1}= {p1}where pn = {wn, wn+1, wn+2, ...1}

The bottom-of-the-scale numeral s is equal to the numeral up to modifies n, so theproposition only denotes a single possibility. This is a violation of the range require-ment, which explains the infelicity of (54-a).

3. The account in degree semantics

Denotation

In the framework of degree semantics, the meaning of DNMs can be formalised as in(55).

(55) Jup toK = �n�P8m 2 [s, ..., n] : P (m) where s > 0 and s 6= n.The degree predicate P holds for all numbers m on a scale from a contextuallydetermined starting point s to the number n; the numeral modified by theDNM. The starting point is higher than 0 and the scale consists of at leasttwo elements (the range requirement).

For instance, (56-a) has the LF in (56-b), using the counting quantifier many (Hackl,2000) as defined in (57). This results in the semantics in (58).

(56) a. Up to 10 people died in the crash.b. [up to 10 [�n [n-many people died in the crash] ] ]

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(57) JmanyK = �n�P�Q.9x[#x = n ^ P (x) ^Q(x)]

(58) 8m 2 [1, ..., 10] : 9x[#x = m ^ people(x) ^ died-in-the-crash(x)]

Ignorance e↵ects

As it is, the meaning of modified numerals with a DNM is now equal to the meaning ofbare numerals, assuming a monotone semantics of bare numerals: up to 10 asserts atleast 10 and implicates no more than 10. With (Nouwen, 2010a), I assume that a morecomplex form that conveys the same meaning as a simple form is blocked (in line withthe Maxim of Brevity, Grice, 1975). To rescue the structure, a speaker possibilityoperator is inserted and the sentence is interpreted with respect to the options thespeaker holds possible. This can be observed in (59).

(59) a. [up to 10 [ ⌃ [ �n [n-many people died in the crash ] ] ] ]b. 8m 2 [1, ..., 10] : ⌃9x[#x = m ^ people(x) ^ died-in-the-crash(x)]

In sum, (56-a) means that the speaker considers it possible that one person died, thattwo people died, ... , and that ten people died.

The insertion of a speaker possibility operator and with the range requirementresult in ignorance e↵ects: since there is always a plurality of numbers on the scale,there will always be multiple possibilities in the speaker’s mind.

Upper-bound implicature

The upper-bound implicature is calculated as follows. If the speaker utters (56-a) whileknowing that 11 ore more people died, the Maxim of Quantity is violated. Hence (60-a)implicates the negation of all alternative propositions with scales ending in numbersabove 10, given in (60-b). The combination of (60-a) and (60-b) lead to the statementin (60-c).

(60) a. 8m 2 [1, ..., 10] : ⌃9x[#x = m ^ people(x) ^ died-in-the-crash(x)] ;b. ¬8m 2 [1, ..., 11] : ⌃9x[#x = m ^ people(x) ^ died-in-the-crash(x)] ^

¬8m 2 [1, ..., 12] : ⌃9x[#x = m ^ people(x) ^ died-in-the-crash(x)]^ ...c. 8m 2 [11, ...,1] : ¬⌃9x[#x = m ^ people(x) ^ died-in-the-crash(x)]

Bottom-of-the-scale e↵ect

As the lower bound s of the scale cannot be zero, the existence of at least one elementfor which the predicate holds is entailed. As shown in (61), the bottom-of-the-scalee↵ect is accounted for.

(61) a. #Up to one person died in the crash.b. [up to 1 [ ⌃ [ �n [n-many people died in the crash ] ] ] ]c. 8m 2 [1, ..., 1] : ⌃9x[#x = m ^ people(x) ^ died-in-the-crash(x)] where

s > 0 and s 6= n.

As s is equal to n, the range requirement is clearly violated, which explains the infelicityof (61-a).

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4. Corpus study

Bottom-of-the-scale e↵ect

I collected some additional data that support the claim that DNMs cannot be combinedwith the number at the bottomof the scale they quantify over from the Corpus ofContemporary American English (COCA) (Davies, 2008).

• I looked at the 29 occurrences of at most one and the first 35 occurrences of upto one.

• There were 23 occurrences of at most one where one is the bottom-of-the-scaleelement, such as in (62-a).

• There were no such cases for up to one. One was not the bottom-of-the-scaleelement in any of the instances of up to one, such as in (62-b).

(62) a. The lesson is that for any group of economic entities to have a unifiedcurrency, there can be at most one independent central bank.6

b. The failure to do so can result in a misdemeanor conviction with punish-ment of up to one year in prison.7

Up to one At most oneOne is the BOTS element 0% 79.3%One is not the BOTS element 100% 20.7%

Table 4: Corpus data on the bottom-of-the-scale e↵ect

Cancellability of the upper bound

A small corpus study I conducted lends further support to the claim that the upperbound of DNMs can be cancelled. I looked for the collocates maybe, perhaps, and evenin the range between one and nine words after up to + number and at most + number.The number of cases of upper bound cancellations I found for each search is given intable 5.8

Up to At mostMaybe 6 0Perhaps 2 0Even 17 0

Table 5: Corpus data on upper bound cancellation

6Source: M. Friedman. (1994). Free-floating anxiety. National Review, 46(17), 32-36.7Source: B. Rankin & G. Bonds Staples. (2011). Sex abuse scandal; Ga. looks to beef up abuse law. Atlanta Journal

Constitution.

8The total number of occurrences of up to + number was 17,586 and the total number of occurrences ot at most +

number was 212. The total number of results for each search were as follows (excluding the cases where up to is not usedas a numeral modifier): up to n; maybe: 14, at most n; maybe: 1, up to n; perhaps: 11, at most n; perhaps: 1, up to n;

even: 80, at most n; even: 2.

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Examples of upper bound cancellations with maybe, perhaps, and even are given in(63) below.

(63) a. We’re talking about up to 20 – or maybe more – $20 billion bled o↵ toSaddam Hussein.9

b. Flood waters of up to 20 feet, perhaps higher, had swept the eastern andsouthern parts of the city seaward of the barricade of wrecked homes anddebris thrown up by the storm.10

c. Bullets stacked in the barrel fire at rates of up to 60,000 rounds perminute, even a million in certain multi-barrel configurations.11

Methodology

I searched for up to one and at most one in the COCA corpus. There were 518occurrences of up to one and 31 occurrences of at most one. Out of the first 100occurrences of up to one, 44 were cases where up to is not used as a numeral modifier(e.g. (64)). I excluded these cases.

(64) The very act of asking opens a guy up to one of his greatest fears: publicdisplays of rejection.12

Out of the other 56 instances, 21 were cases where one is part of a higher number ora fraction, as in (65).

(65) Up to one hundred twenty women will participate in the trials and they’redefinitely not doing it for the money.13

The remaining 35 were cases where one is not the bottom-of-the-scale element, as in(66).

(66) The failure to do so can result in a misdemeanor conviction with punishmentof up to one year in prison.14

Table 6 summarises these results. In the table in section 3, I disregarded the caseswere one is part of a higher number or fraction.

9Source: Interview with Je↵ Flake, Neil Cavuto, Fox, 2004.10Source: P. Hughes. (1990). The great Galveston hurricane. Weatherwise, 43(4).11Source: E. Adams. (2004). Is This What War Will Come To? Even as the Pentagon struggles with the low-tech

reality of war in Iraq, it looks to increasingly bizarre-sounding technology for next-gen fighting systems. On the followingpages, five chapters from the Pentagon’s sci-fi future. Popular Science, 264(6).

12Source: Ask him anything, Cosmopolitan magazine, September 2008.13Source: CBS News, 28 September 2009.14Source: B. Rankin & G. Bonds Staples. (2011). Sex abuse scandal; Ga. looks to beef up abuse law. Atlanta Journal

Constitution.

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Up to one At most oneHigher number/fraction (e.g. (65)) 21 2One is the BOTS element 0 6One is not the BOTS element 35 23

Total 56 31

Table 6: Corpus data on up to one and at most one

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